The synchronicity of dynamic epistemic logic
Cédric Dégremont
Afdeling Kunstmatige Intelligentie
Rjiksuniversiteit Groningen
Bernoulliborg
Nijenborgh 9
9747 AG Groningen
The Netherlands
Benedikt Löwe
Institute for Logic, Language
and Computation
Universiteit van Amsterdam
Postbus 94242
1090 GE Amsterdam
The Netherlands
Abstract
Van Benthem, Gerbrandy, Hoshi and Pacuit
gave a natural translation of dynamic epistemic logic (DEL) into epistemic temporal logic (ETL) and proved a representation
theorem, characterizing those ETL models
that are translations of some DEL protocol;
among the characterizing properties we also
find synchronicity. In this paper, we argue
that synchronicity is not an inherent property
of DEL, but rather of the translation that van
Benthem et al. used. We provide a different
translation that produces asynchronous ETL
models and discuss a minimal temporal extension of DEL that removes the ambiguities
between the possible translations. This allows us a first attempt of an assessment which
of the epistemic-temporal properties are intrinsic to DEL and which are properties of
the translation.
1
Introduction
Dynamic epistemic logic (DEL) (Baltag et al., 1998),
while being about changes in epistemic status over
time, is not a temporal logic and cannot itself express
temporal aspects. In order to model the temporal aspects of epistemic and doxastic change represented in
dynamic epistemic logic, we need to embed DEL into
some epistemic temporal system such as ETL (Parikh
and Ramanujam, 2003) or Interpreted Systems (Fagin
et al., 1995). ETL is a close cognate to DEL in the
sense that its also events do not include a notion of
agency or turns, as opposed to some other multi-agent
epistemic-temporal logics such as ATEL (van der Hoek
and Wooldridge, 2003).
Representation theorems linking DEL and ETL were
proposed for the first time in (van Benthem, 2001).
Recently, van Benthem, Gerbrandy, Hoshi, and Pacuit
Andreas Witzel
NYU Bioinformatics Lab
Courant Institute for the
Mathematical Sciences
New York University
New York, NY 10003
United States of America
2009 gave a natural model-based translation from the
framework of DEL into ETL: for each DEL protocol,
they produce an ETL forest generated by sequentially
updating an epistemic model using the DEL product
update. They then showed that the temporal translation of DEL thus obtained always produces ETL models satisfying a number of properties, including synchronicity and perfect recall. We shall refer to this
result (precisely given as Theorem 6) by “the vBGHP
representation”.
The vBGHP representation has been interpreted to
imply that DEL can inherently only model agents that
are synchronous.1 Our aim here is to stress that this is
an oversimplification of the mentioned result; in fact,
synchronicity is a property of the described translation, not a property of the logic. Other translations of
DEL into ETL might very well exhibit other features.
Our aim is to clarify the meaning of the vBGHP representation, with the long-term goal to identify those
properties that do not depend on the choice of embedding of DEL and ETL and those which do.
The paper is organized as follows: After giving the
general framework (§ 2) and discussing the concept of
perfect recall (§ 3), we propose an alternative way of
embedding DEL into ETL, exhibiting a natural example of an asynchronous ETL model that arguably represents faithfully the dynamic component of DEL (Example 13), and finally state a representation theorem
for our embedding (§ 4). Comparing the properties
characterizing asynchronously DEL generated forests
(Theorem 16) to the properties used in the vBGHP
representation (Corollary 9), we discuss which properties might be regarded as core DEL properties (§ 5).
Finally, we then generalize the two interpretations by
introducing clock tick functions which allows to cover
a whole range, having the two given constructions as
1
E.g., (Renne et al., 2009, p. 1): “[van Benthem et al.]
showed that standard Dynamic Epistemic Logic necessarily
satisfies synchronicity”.
extremal special cases (§ 6). In § 7, we note that the
cases of asynchronicity are rather limited if we restrict
our attention to S5 and sketch some avenues for future
research, including an investigation of sub-S5 settings.
Related Work.
Our work fits in the research tradition at the interface between DEL and ETL which is, e.g., studied in
(Sack, 2007) and (Hoshi, 2009) where the reader can
find logics designed at this interface, merging the dynamic and the temporal paradigms. More results in
this direction can be found in (Wang et al., 2010) and
its unpublished extended version. The vBGHP representation (the main motivation of the present paper)
is published in (van Benthem et al., 2009) together
with other results about the merging of the dynamic
and the temporal framework. A combination of temporal logic and dynamic epistemic logic can be found
in (Renne et al., 2009).
In this paper, our interest is not so much the merging of the frameworks and the design of logics that
have both dynamic and temporal aspects, but rather
the study of the translation between the dynamic and
the temporal framework. In this study, we encounter
natural temporal properties such as synchronicity and
perfect recall. These properties have been studied
by other authors. E.g., Renne et al. (2009) extend
basic DEL models and protocols to account for nonsynchronous scenarios and Isaac and Hoshi (to appear)
discuss transformations of asynchronous models into
synchronous models (thus resolving the diachronic uncertainty).
In our investigation of synchronicity of the translation
used in the vBGHP representation, we notice that the
notion of perfect recall used in the theorem in a sense
presupposes synchronicity. This lead to a separate
study of different notions of perfect recall in (Witzel,
2011). We shall use its analysis in our § 3, and consider
it a sibling to the present paper.
Finally, in § 7, we discuss possible directions of future
work such as weakening the S5 assumptions or adding
more structure to the models. Papers relating to the
research directions are listed and discussed in § 7.
2
2.1
Basic definitions and the vBGHP
representation
DEL and ETL models and their
properties
In the following, we shall deal with various types of relational structures, building on the same sets of propositional variables and agents that we denote by prop
and N , respectively. Modal languages that can be interpreted on such structures could have three types of
modalities: epistemic, temporal and action modalities.
An (S5) epistemic model M = (W, (∼i )i∈N , V ) consists of a nonempty set W of worlds, an equivalence
relation ∼i for every agent, and a valuation function
V : prop → ℘(W ). We also write |M| for W . A frame
is a model without valuation function. Event models
are triples E = hE, (∼Ei )i∈N , prei, where E 6= ∅ is a
set of events, for each agent i ∈ N , ∼Ei is an equivalence relation on E, and pre : E → L is a precondition
function mapping events into some epistemic language
L. As usual, a pointed event model is an event model
with one distinguished element from E.
We shall not give an introduction to dynamic epistemic
logic here, but rather refer the reader to the textbook
(van Ditmarsch et al., 2007). The crucial operation for
DEL is that of the product update:
Definition 1 (Product Update). The product update
of an epistemic model M = hW, (∼i )i∈N , V i with an
event model E = hE, (∼Ei )i∈N , prei is the model M⊗E
whose states are the pairs (w, e) such that w satisfies
the precondition of the event e and whose epistemic
relations are defined as:
(w, e) ∼0i (w0 , e0 ) iff e ∼Ei e0 , w ∼i w0
and whose valuation is defined by
(w, e) ∈ V 0 (p) iff w ∈ V (p), for all p ∈ prop.
We turn to structures giving a temporal perspective on
information change: an (S5) epistemic temporal model
or (S5) ETL model H is a tuple hΣ, H, (∼i )i∈N , V i
with Σ a finite set of events, and H ⊆ Σ∗ closed under
non-empty prefixes (i.e., is a forest of events). For each
i ∈ N , ∼i is an equivalence relation on H, and there
is a valuation V : prop → ℘(H).
We introduce some notation for ETL models that we
shall use later. In the following, we fix an ETL model
H = hΣ, H, (∼i )i∈N , V i.
For any sequence σ, let σ[n] be the nth element of σ.
We write h ≤ h0 iff there exists some (possibly empty)
sequence of events σ ∈ Σ∗ such that h0 = hσ, and write
Ki [h] = {h0 | h ∼i h0 }. Let Hn = {h | len(h) = n} and
H≤n = {h | len(h) ≤ n}. Given an ETL model H and
m < n, H \ (m, n)
S is the restriction of the relations in
H to (H × H) \ 1≤i≤m ((Hi × Hn ) ∪ (Hn × Hi )).
Definition 2. For an agent i and two pointed models M, w and M0 , w0 , we say that w and w0 are ibisimilar, in symbols w 'i w0 , iff
1. for each v ∈ M with w ∼i v there is v 0 ∈ M0 such
that w0 ∼i v 0 and v ' v 0
2. for each v 0 ∈ M with w0 ∼i v 0 there is v ∈ M
such that w ∼i v and v ' v 0 ,
where ' denotes the standard epistemic bisimulation.
Because in this paper the role of preconditions will
be trivialized by state-dependent protocols (Definition
3), we can proceed and state representation theorems
independently of a choice of epistemic language. If we
fix an epistemic language and a pointed model M, w,
we could formulate a notion of theory of agent i at
w consisting of all statements of the form Ki ϕ and
¬Ki ϕ true at w (where Ki is the knowledge operator
for agent i). On finite frames, being i-bisimilar and
having the same i-theory are equivalent.
Given two pointed ETL models H, h, H0 , h0 and m <
\(m,n)
n, we write that H, h 'i
H0 , h0 iff H\(m, n), h 'i
H\(m,n) 0
H0 \ (m, n), h0 . We sometimes write h 'i
h to
\(m,n)
mean H, h 'i
H, h0 .
As a last concept to be introduced, if (e1 . . . en ) ∈ H
is a history, we define agent i’s experience record to be
the sequence
EEi (e1 . . . en ) := Ki [e1 ]Ki [e1 e2 ] . . . Ki [e1 . . . en ],
and write EEi (h1 ) ≈i EEi (h2 ) iff EEi (h1 ) is equivalent
to EEi (h2 ) up to stuttering.
In the following, we define a number of properties of
ETL frames that will play an important role in our
representation theorems.
We say that an S5 ETL model H = hΣ, H, (∼i )i∈N , V i
satisfies
0
• Synchronicity (Syn) iff for all i, h, h with h ∼i
h0 we have len(h) = len(h0 )
• Perfect recall (PR) iff for all i, h, e, h0 with he ∼i
h0 there is some h00 ≤ h0 such that h ∼i h00 .
• Weak synchronous perfect recall (wsPR) iff
for all h, h0 , e, e0 with he ∼ h0 e0 , we have h ∼ h0 .
• Synchronous perfect recall (sPR) iff for all
i, h, h0 , e, e0 with he ∼i h0 there exists some e0 and
h00 such that h0 = h00 e0 and h ∼i h00 .
• Synchronous grounding (SG) iff for all i, h, h0
with h ∼i h0 and len(h) ≤ len(h0 ), there is some
h00 ≤ h0 with len(h) = len(h00 ) and h ∼i h00 .
• (Synchronous) uniform no miracles (UNM)
iff for all i, if there are h, h0 , e, f with len(h) =
len(h0 ) and he ∼i h0 f , then for all g, g 0 such that
g ∼i g 0 and len(g) = len(g 0 ) we have ge ∼i g 0 f .
• Weak uniform no miracles (wUNM) iff for all i,
if there are h, h0 , e, f with len(h) = len(h0 ), he ∼i
h0 f and h 6∼i he, then for all g, g 0 such that g ∼i g 0
and len(g) = len(g 0 ) we have ge ∼i g 0 f .
• Perfect tracking (PT) iff for all i, h, e, n, if h ∼i
H\(n,n+1)
he and len(h) = n then h 'i
he.
• No pure time perception (NPTP) iff for all
H\(n,n+1)
i, h, e, n, if h 'i
he and len(h) = n then
h ∼i he.
• Propositional stability (PS) iff for all h, h0 and
all p ∈ prop, if h ≤ h0 then h ∈ V (p) iff h0 ∈ V (p).
2.2
The vBGHP representation
We now give an account of the vBGHP representation.
(Liu, 2008, Chapter 5) surveys earlier representation
results. For a less compact presentation of the vBHGP
representation the reader should consult (van Benthem
et al., 2009, § 3).
Let E be the class of all pointed event models; we let
P(E) := {P ⊆ E∗ |P is closed under finite prefixes}
be the set of all forests of sequences of event models.
Definition 3. Let M be an epistemic model. A statedependent protocol for M is a mapping P from |M| to
P(E).
S
We define Pσ [M] =
Furthermore,
v∈|M| P(v).
Pe [M] = {(E, e) | (E, e) occurs in some σ ∗ ∈ Pσ [M]}.
The following two definitions are the foundations of
the vBGHP representation:
Definition 4 (P-generated model, (van Benthem
et al., 2009)). Let M = (W, (∼i )i∈N , V ) be an epistemic model and P a state-dependent protocol for
M. The P-generated model at level n, Mn,P =
n,P
hW n,P , (∼n,P
i is inductively defined as foli )i∈N , V
lows:
• M0,P = M,
• W n+1,P = {wσ(E, e) | wσ ∈ W n,P , σ(E, e) ∈
P(w) and Mn,P , wσ pre(E, e)},
• for every wσ(E, e), vσ 0 (E 0 , e0 ) ∈ W n+1,P we set
wσ(E, e)∼n+1,P
vσ 0 (E 0 , e0 ) iff E = E 0 , e ∼Ei e0
i
n,P
wσ ∼i vσ 0 ,
• and, for every wσ(E, e) ∈ W n+1,P and p ∈ prop,
we set wσ(E, e) ∈ V n+1,P (p) iff wσ ∈ V n,P (p).
Definition 5 (Synchronously DEL generated ETL
models, (van Benthem et al., 2009)). Let M =
(W, (∼i )i∈N , V ) be an epistemic model and P be a
state-dependent DEL protocol for M, the ETL model
synchronously generated by M and P is defined as
sForest(M, P) := (Σ, H, {∼i }i∈N , V 0 ),
where
1. Σ = Pe [M] ∪ |M|,
2. H =
S
n≥0
W n,P ,
0
0
3. for all h, h ∈ H ∩ W , h ∼ h iff h
∼0,P
i
0
h.
4. for all h, h0 ∈ H \ W with h = wσ and h0 = vσ 0 ,
len(σ),P 0
h ∼i h0 iff len(σ) = len(σ 0 ) and h ∼i
h,
5. and, for all p ∈ prop, h ∈ V 0 (p) iff h ∈
V len(h−1),P (p).
Note that we denote the translation function from (van
Benthem et al., 2009) by sForest (for “synchronously
DEL generated ETL Forest”) in order to distinguish it
from our later notion of asForest (for “asynchronously
DEL generated ETL Forest”). Using Definitions 4 and
5, we can now state the vBGHP representation:
Theorem 6 (van Benthem, Gerbrandy, Hoshi,
Pacuit). For an S5 epistemic-temporal ETL model,
the following are equivalent:
1. H is isomorphic to sForest(M, P) for some epistemic model M and some state-dependent DEL
protocol P, and
2. H satisfies wsPR, Syn, UNM, and PS.
3
The issue of perfect recall
The reader might wonder why we have three different
notions of perfect recall in our list of properties of ETL
models: PR, sPR and wsPR. The common formulation
of perfect recall from the interpreted systems literature is our PR (Fagin et al., 1995).2 It is however, not
the definition of perfect recall in the vBGHP representation. As can be seen in Theorem 6, the authors of
van Benthem et al. (2009) use wsPR instead.3
Observation 7. On synchronous S5 ETL models, PR,
wsPR, and sPR are equivalent.
Proof. Cf. (Witzel, 2011).
2
More precisely, it is a slight and equivalent variant of
the formulation in (Halpern et al., 2004, Lemma 2.2(d)).
3
sPR was the notion of perfect recall used in the earlier
representation results (van Benthem, 2001, 2006).
However, in general, PR does not imply wsPR (see below, Proposition 11). While PR has intuitive interpretations (Witzel, 2011, where PR is called PRhc ), wsPR
is used as a technical notion without intuitive motivation; only if one presupposes synchronicity, wsPR
inherits the intuition from PR. For this reason, we use
PR as the fundamental definition of perfect recall.
Observation 8. On synchronous S5 ETL models,
UNM and wUNM are equivalent.
Using Observations 7 and 8 and the fact that SG and
PT are trivially true on synchronous models, we can
now restate the vBGHP representation in terms of PR.
Corollary 9. For an S5 epistemic-temporal ETL
model, the following are equivalent:
1. H is isomorphic to sForest (M,P) for some epistemic model M and some state-dependent DEL
protocol P, and
2. H satisfies PR, Syn, PT, SG, wUNM, and PS.
We end this section by introducing two equivalent alternative characterizations of PR, which will be convenient for some of our later proofs. The first one is
a natural notion of perfect recall very close in spirit
to the game-theoretic one (Osborne and Rubinstein,
1994, Chapter 11).
Definition 10. We say that an ETL model H =
hΣ, H, (∼i )i∈N , V i satisfies
• perfect recall based on epistemic experience (PRee ) iff for all i, h, h0 with h ∼i h0 , we
have EEi (h) ≈i EEi (h0 ).
• perfect recall, local version (PR` ) iff for each
i, h, h0 , e with he ∼i h0 , one of the following holds:
(i) h ∼i h0
(ii) h ∼i h00
(iii) he ∼i h00
where h00 is the direct predecessor of h0 , i.e., h0 =
h00 e0 for some e0 .
The following result states the relationships between
the various versions of perfect recall:
Proposition 11. On S5 ETL models, PR, PRee and
PR` are equivalent; wsPR neither implies nor is implied
by these; and sPR is equivalent to synchronicity plus
PR. On synchronous S5 ETL models, all of the notions
are equivalent.
Proof. Cf. (Witzel, 2011). Figure 1 is also taken from
(Witzel, 2011) and provides the examples that show
that PR and wsPR do not imply each other (the relevant part of the claim of Proposition 11 for our present
purposes).
e3
e3
e1
e2
e1
(a) PR, but not wsPR
e2
(b) wsPR, but not
PR
Figure 1: Two S5 ETL models, gray lines indicating
information sets (reflexive loops omitted). (a) PR does
not imply wsPR (wsPR is violated because e1 e3 ∼ e2 e3
but e1 6∼ e2 ), and (b) wsPR does not imply PR on ETL
forests: Intuitively, event e1 lets the agent “forget”
that he is in the left tree.
all this is common knowledge). The event model E
with domain {(e1 , e2 )}, with ∼j being the minimal reflexive relation and ∼i the universal relation, and with
pre(e1 ) = p and pre(e2 ) = > represents the event in
which agent j learns that p is true (e1 ), but agent i
considers it possible that nothing happens (e2 ). In
the resulting model M ⊗ E, j knows p, while agent i’s
knowledge has not changed, since he already before
considered it possible that j actually knew p. All of
this is represented in Figure 2.
>
¬p
w1
¬p
w2
i
w2 e2
i
Asynchronous ETL models and our
representation theorem
In this section, we give our alternative translation from
DEL to ETL that will in general produce asynchronous
ETL models.
Definition 12 (Asynchronously DEL generated ETL
models). Let M = (W, (∼i )i∈N , V ) be an epistemic
model and P be a state-dependent DEL protocol for
M, the ETL model asynchronously generated by M
and P is defined as
asForest(M, P) := (Σ, H, {∼0i }i∈N , V ),
where (Σ, H, {∼i }i∈N , V ) = sForest(M, P), and where
∼0i is the symmetric transitive closure of
w1 e1
¬p
¬p
i, j
i
w2 e1
i
p
w1 e2
i
i
sForest(M,P)
=
e1
=
i
p
w 3 e1
w 2 e2
w4 e1
i
¬p
i, j
p
w3 e2
i
p
w4 e1
i
w 4 e2
Figure 2: DEL model M, event model E, and resulting
model M ⊗ E, first shown to illustrate the product
operation and then transformed into a line. Transitive
and reflexive accessibilities omitted.
For any w ∈ |M|, let P(w) = {(E, e1 ), (E, e2 )}. We
now apply sForest to obtain an ETL model as given
in Figure 3. By design, sForest(M, P) just consists of
M ⊗ E stacked on top of M.
ii
∼i ∪{(wσ, wσ(E, e)) ∈ (H × H) | σ(E, e) ∈ P(w)
and wσ 'i
i
p
w4 e2
i
p
w2 e1
e2
p
w 3 e2
¬p, p
i, j
⊗
w4
i
i
p
¬p, p
p
w3
i, j
p
w1 e2
i
4
¬p
p
jj
wσ(E, e)}.
(asForest stands for “asynchronously DEL generated
ETL Forest”.)
To get a feeling for the definition of asForest, let us
consider the simplest possible example: an epistemic
model M consisting of one world w, and the trivial
event model E consisting of one event e with precondition >. Then M and M ⊗ E are isomorphic, thus
bisimilar, and therefore w 'i we. In the ETL model
asForest(M, P), we have two elements, w and we, and
we know that w ∼0i we, violating synchronicity.
We now give a more elaborate example:
Example 13. The model M in Figure 2 represents
a situation with two agents i, j in which neither agent
knows the truth value of a proposition p, but i considers it possible that j does know either p or ¬p (and
e1
ii
e1
e2
e1
e1
e2
jj
Figure 3: sForest(M, P)
If we inspect the epistemic model, we realize that
w2 'i w2 e2 , w3 'i w3 e1 , w2 'j w2 e2 , and w3 'j
w3 e1 . Using this fact for the definition of the equivalence relations ∼0i and ∼0j , we obtain the ETL model
given in Figure 4.
The event E represents that j gains some information,
but i does not know whether j gains this information; it does not specify whether i does not realize
that anything is happening at all. Whether this is
the case, depends on the situation that is being modelled: if i sees j talk to some other agent who knows
ii
jj
e1
e1
e2
e1
e1
e2
Figure 4: asForest(M, P)
whether p, but cannot hear the conversation, one may
argue that i does realize that something is happening even though his epistemic state as represented in
the model is not changing; in this case sForest is more
appropriate. On the other hand, if i cannot see this
conversation, then asForest represents this state of affairs better than sForest.
The event model itself does not distinguish between
these two intended interpretations. In Section 6, we
therefore propose a slight extension of DEL models
that will allow for this distinction.4
As a first property of our proposed translation, we
prove that asForest satisfies perfect recall.
Proposition 14. For any model M and DEL protocol
P, asForest(M, P) has perfect recall.
Proof. First we note that sForest(M, P) has perfect
recall. For any two histories h1 , h2 and events e1 , e2 ,
the product update allows h1 e1 ∼ h2 e2 only if h1 ∼ h2 ,
so PR is satisfied.
The rest of the proof uses the (equivalent) definition
PRee for perfect recall. That is, we show that in any
asForest, whenever h1 ∼i h2 , then EE(h1 ) ≈i EE(h2 ).
We start by showing that equivalence of epistemic experiences modulo stutterings is preserved under joining information sets. Formally, for any agent i, consider any two histories h1 , h2 of some forest with accessibility relation ∼i which has EEi (h1 ) ≈i EEi (h2 ).
Then in the forest that is obtained by letting h01 ∼i h02
for any two histories h01 , h02 and closing ∼i off under
symmetry and transitivity, we still have EEi (h1 ) ≈i
EEi (h2 ).
To see this, first recall that i’s epistemic experience
EEi (h) at some history h is the sequence of information
4
The distinction between internal and external perspective (Aucher, 2008) becomes relevant here: if we view DEL
as “internal reasoning engine” of some (artificial) agent,
the fact that that agent even updates the current model
means that he was notified in some way about the event,
even if the update has no effect on his represented epistemic state; in that case, only sForest seems adequate. If
we assume the viewpoint of an external modeler who keeps
track of agents’ epistemic states as events occur, asForest
becomes plausible.
sets he has gone through. Joining the two information
sets [h01 ]∼i and [h02 ]∼i results in replacing occurrences
of either of these two sets in that sequence by one
and the same set [h01 ]∼i ∪ [h02 ]∼i . The relation ≈i ,
equivalence modulo stutterings, can thus only grow
when joining information sets.
We now turn to the main claim. We show that adding
a single “vertical” accessibility h ∼i he for some history h and event e and closing off under symmetry and
transitivity preserves perfect recall. The main claim
then follows inductively since asForest(M, P) is obtained from sForest(M, P) by several such operations
and, as noted above, sForest(M, P) has perfect recall.
So let H denote some S5 ETL model with perfect
recall, h and he some histories in H, and H0 the
ETL model with h ∼i he added and closed off. For
any h1 , h2 with h1 ∼i h2 already in H, we have
EEi (h1 ) ≈i EEi (h2 ) in H since H has perfect recall. As established above, the relation ≈i can only
grow when joining information sets, and so we have
EEi (h1 ) ≈i EEi (h2 ) also in H0 .
It remains to show that the condition of PRee is satisfied for the accessibilities added in F 0 . For the explicitly added accessibility h ∼i he, we have [h]∼i = [he]∼i
in F 0 and thus EEi (h) ≈i EEi (he) since ≈i disregards
stutterings. For the accessibilities added by closing
off under symmetry and transitivity, the condition is
satisfied by symmetry and transitivity of ≈i .
Lemma 15. For any ETL model H with perfect recall
and synchronous grounding, and for any two histories
he and h0 e0 , if len(h) = len(h0 ) and he ∼ h0 e0 then
h ∼ h0 .
Proof. Since PR` characterizes perfect recall, we obtain one of these three cases:
(i) h ∼ h0 e0 , then h ∼ h0 follows from synchronous
grounding;
(ii) h ∼ h0 , done;
(iii) he ∼ h0 , then h ∼ h0 again follows from synchronous grounding.
We are finally ready to state our main theorem.
Theorem 16. For an S5 epistemic-temporal ETL
model, the following are equivalent:
1. H is isomorphic to asForest(M, P) for some epistemic model M and some state-dependent DEL
protocol P, and
2. H satisfies PR, NPTP, PT, SG, wUNM, and PS.
Compare this to Corollary 9 where sForest is characterized in terms of the same properties, except that
NPTP is replaced by Syn. The proof can be found in
Appendix A.
5
Relating the two representation
theorems
We should stress that there is no conflict between the
two representation theorems (the two embeddings are
different, hence so are the epistemic temporal properties), and that none of them is a generalization of
the other. The two translations sForest or asForest are
opposite extremes of interpreting the ambiguity in the
process of translating temporally underspecified DEL
models into ETL models.
Exploring these two extremes can help us to understand better which of the temporal properties we investigated are core DEL properties and which ones are
properties that depend on the choice of the embedding. Properties shared by the two embeddings are
candidates for core DEL properties. We briefly discuss the various properties involved, discussing their
degree of entrenchment in DEL, their desirability (for
flexible formalisms for modelling multi-agent systems),
and possibilities of lifting them. Of course, questions
of desirability as well as questions of whether a given
embedding is natural are non-mathematical questions
and will not be answered definitively.
Synchronicity. As demonstrated through the two
translations, DEL can be interpreted as either satisfying Syn or NPTP. These are diametrically opposed properties of a range of conceivable perceptions of time. Section 6 shows how DEL can be
extended to explicitly distinguish between these
two extremes and the whole range between them.
Perfect recall is a consequence of the way the DEL
product update works: intuitively, it can only remove epistemic accessibilities. With respect to
what was the case before the event happened,
product update can only reduce uncertainty, thus
can only increase knowledge. This monotonicity
of informational states, under new update by new
events, is precisely what doxastic versions of DEL
using priority update (Baltag and Smets, 2006;
van Benthem, 2007) weaken, enabling agents to
truly revise their beliefs (cf. § 7).
Synchronous grounding reflects the fact that
product update is applied uniformly to the whole
model. This seems rather deeply entrenched
in DEL, and defining updates that are only
applied to parts of a model seems difficult. An
interpretation of the property is that if Alice
considers possible some history h that she is no
longer synchronous with, it must because when
she was still synchronous with it, she was already
considering h possible.
Weak uniform no miracles (like its stronger version) reflects the fact that product update is a
local, i.e. history-independent, update.
Perfect tracking. The fact that successive states of
the world can only be indistinguishable if an
agent’s epistemic state has not changed is natural
and justified: it is analogous to standard epistemic introspection assumptions.
Propositional stability is (as is well-known) a serious restriction and can be removed by considering ontic actions (see, e.g.,(van Benthem et al.,
2006)).
6
A minimal temporal extension of
DEL
A natural question is thus how to enable DEL to distinguish between classes of ETL models assuming the
mentioned extremal perceptions of time, and any intermediate ones. We can do so using a very simple idea:
a history-dependent clock ticking function. This minimalistic extension of protocols-based DEL removes the
temporal ambiguity and enables DEL to distinguish,
at any point in time, between the asynchronous and
the synchronous interpretation.5
Our proposed extension enriches DEL protocols with
flags, one for each agent, specifying whether or not a
new epistemic situation is (potentially) indistinguishable from the old situation for that specific agent. Intuitively, this can be thought of as specifying whether
or not an agent hears a clock tick as this event occurs. In terms of epistemic experience, this flag specifies whether a stuttering resulting from this atomic
event should be conflated or not.
We make this idea precise in the following sequence of
definitions.
Definition 17. Given a state dependent protocol P
for M a history dependent clock tick function is a
function c : (|M| × Pσ [M] × E) → 2N .
E.g., we interpret c(w, σ, (E, e)) = {i} to mean
5
Of course, more sophisticated extensions of DEL that
account for synchronous and asynchronous scenarios are
possible, as the one proposed by Renne et al. (2009). We
are aiming here for the minimal extension that does the
job.
if the initial state was w, the sequence σ of
events has occurred, and (E, e) happens, then
agent i, even if his or her epistemic state remains the same, will notice that time has
passed; and no one else does.
Now we turn to the construction.
Definition 18 (clock tick function DEL generated
ETL models). Let M = (W, (∼i )i∈N , V ) be an epistemic model and P be a state-dependent DEL protocol
with a clock tick function c, the ETL model generated
by M, P and c is defined as
clickForest(M, P, c) := (Σ, H, {∼0i }i∈N , V ),
where (Σ, H, {∼i }i∈N , V ) = sForest(M, P), and where
∼0i is the symmetric transitive closure of
∼i ∪{ (wσ, wσ(E, e)) ∈ (H × H) |
σ(E, e) ∈ P(w)
sForest(M,P)
and wσ 'i
wσ(E, e)
and i 6∈ c(w, σ, (E, e))}.
(clickForest stands for “clock tick function DEL genrated ETL Forest”.)
We now give a representation theorem that accounts
for the whole class of ETL models having the mentioned DEL-originated epistemic-temporal properties.
Theorem 19. For an S5 epistemic-temporal ETL
model, the following are equivalent:
1. H is isomorphic to clickForest(M, P, c) for some
epistemic model M, some state-dependent DEL
protocol P and some clock tick function c.
2. H satisfies perfect recall, perfect tracking, synchronous grounding, synchronous uniform no miracles and propositional stability.
The proof can be found in Appendix B.
Examples. The empty event (one atomic event e with
precondition >) can now be used to represent events of
pure (loss of) synchronicity among agents, depending
on the clock tick function c. If c(e) = A ⊆ N , the event
has the effect that all agents in A stay “in sync”, while
all others get “out of sync”.
Another practical use of this extension is for incorporating a form of agency into DEL events, bringing it
closer in spirit to game theoretic or autonomous agent
frameworks. In those frameworks, events are predominantly taken to be a particular agent’s actions, and
the actions performed by an agent are assumed to be
part of the local state description (in case of perfect
recall agents), so that agents can always distinguish
between states of the world in which the sequence of
their own actions differs. We can now specify which
(atomic or composite) events constitute an agent’s set
of actions by setting the flag so that he “hears a clock
tick” whenever he performs one of his actions, thus
ensuring that he can discern whether and how he has
acted even if his epistemic state (in terms of the DEL
model) doesn’t change.
7
Conclusions and future work
The constructions given in Definition 12 (without clock
tick function) and Definition 18 (with clock tick function) can generate asynchronous models from statedependent DEL protocols. But it should be noted that
the diachronic uncertainties generated this way are of
a very special type. In particular, (non-trivial) asynchronicities are only obtained if an agent already “happens to” have anticipated the possibility of an event
occurring. Put differently, there is no (non-trivial)
event that inherently gives rise to asynchronicity; it
always depends on a specific situation.
This observation is not unexpected; after all, we are
dealing with S5 knowledge: Agents cannot be mistaken, so if an event occurs that they do not notice,
they must already have considered its consequences
possible. However, for modeling realistic multi-agent
systems in which agents may be really separate entities and make private observations, this is a severe
restriction.
Future work. To account for a wider ranger of
diachronic uncertainties and more general situations,
two natural directions can be explored. The first one
is to give up S5, i.e., weaken the assumption that
epistemic accessibility relations are equivalence relations. The second is to consider richer models that include both an epistemic accessibility relation, encoding
agents’ information, and a plausibility ordering, indicating which states (or histories) are a priori more
likely, taking beliefs to be defined as ‘true in the most
plausible states in the current information set’.
In the first direction it might be the case that some
of the previous properties would need to be adjusted,
were we to drop the assumptions that we are working with equivalence relations, i.e., with partitions.
It is however difficult to make any conjecture, since
the S5 assumption plays a role in numerous places in
our proofs. Moreover the natural symmetry with the
synchronous case would probably be lost. As an example, recall from Proposition 11 that if we drop the
S5 assumption different notions of perfect recall start
to become incomparable. This contrasts drastically
with the situation in the synchronous case: indeed the
vBGHP representation do not require any of the transitivity, reflexivity or symmetry assumption.
Oliver Board. Dynamic interactive epistemology.
Games and Economic Behavior, 49:49–80, 2004.
In the second direction, epistemic models are enriched
into epistemic-plausibility models (Board, 2004; van
Ditmarsch, 2005; Baltag and Smets, 2006; van Benthem, 2007). Similarly ETL models can be enriched
and this has lead to similar synchronous representation results (van Benthem and Dégremont, 2010;
Dégremont, 2010). Now, it is not immediate to decide what would be the most natural way to extend
the idea behind our asynchronous construction to this
richer setting. Indeed, the intuition of the asychronous
construction was that “agents can only know that
some event has happened, if their epistemic state has
changed”. We could adapt it as follow “agents can
only believe that some events happened, if their doxastic state has changed”, leading to the idea that by
default, if nothing has change in the doxastic state
of the agent after some event, then she should still
consider it most plausible that nothing has happened.
This could account for scenarios in which agents have
opposite beliefs about whether some event has happened or not.
Cédric Dégremont. The Temporal Mind. Observations on the logic of belief change in interactive
systems. PhD thesis, Universiteit van Amsterdam,
2010. ILLC Publications DS-2010-03.
Acknowledgements
The research of the first and second author was partially supported by the Early Stage Research Training Mono-Host Fellowship GLoRiClass funded by the
European Commission (MEST-CT-2005-020841). This
project started while the second author was visiting
the CUNY Graduate Center in New York in the fall of
2009; he expresses his gratitude for the hospitality of
his hosts.
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Appendix
Appendix
A
Proof of Theorem 16.
A piece of notation: given an ETL model H, H|n is
the restriction of H to Hn , H|≤n is the restriction of
H to H≤n .
Observation 20. Let H = hΣ, H, (∼i )i∈N , V i be an
S5 ETL model and let (he, h0 f ) ∈ Hn+1 × Hn+1 be
\(n,n+1)
such that h ∼i h0 , and Hk , h 'i
Hk , he and
\(n,n+1)
\(n,n+1)
Hk , h0 'i
Hk , h0 f . We have Hk , he 'i
Hk , h0 f .
Definition 21. Let H = hΣ, H, (∼i )i∈N , V i and H0 =
hΣ0 , H 0 , (∼0i )i∈N , V 0 i be two S5 ETL models. We say
that H0 is n + 1-bisimulation closure reachable from H
iff the following conditions hold:
1. Σ = Σ0 ; H = H 0 ; for every h ∈ H and p ∈ prop
h ∈ V (p) iff h ∈ V 0 (p)
2. for every i ∈ N and (h, h0 ) ∈ (H × H) \ (Hn+1 ×
Hn+1 ) we have h ∼i h0 iff h ∼0i h0
3. for every i ∈ N , ∼i ⊆ ∼0i
4. there exists a sequence of ETL models H0 . . . Hm
with H0 = H, Hm = H0 such that for all k with
0 ≤ k < m, we have Hk = hΣ, H, (∼ki )i∈N , V i
and there exists i ∈ N and there exists a
pair of histories (he, h0 f ) ∈ Hn+1 × Hn+1 with
\(n,n+1)
Hk , he 'i
Hk , h0 f such that ∼k+1
is the
i
reflexive, transitive and symmetric closure of ∼ki
∪{(he, h0 f )} and for every j ∈ N \ {i}, ∼k+1
=∼kj .
j
The following is a general fact about S5 ETL models,
that will be useful in this paper.
Fact 22. Let H = hΣ, H, (∼i )i∈N , V i and H0 =
hΣ0 , H 0 , (∼0i )i∈N , V 0 i be two S5 ETL models with H0
n + 1-bisimulation closure reachable from H. We
\(n,n+1)
claim that for all j ∈ N , H, g 'j
H, ga iff
H0 , g 'j
\(n,n+1)
H0 , ga.
Proof. The proof is by induction on the length of the
witness sequence for n + 1-bisimulation closure reachable from H. The base case is bisimilarity of identical structure. Now assume the claim holds for sequences of length k. Now, assume that i is the witness agent and (he, h0 f ) the witness pair to obtain
Hk+1 from Hk . From the definition of the witness
\(n,n+1)
sequence we have Hk , he 'i
Hk , h0 f . By inspection of the clause of a j-bisimulation, we can see
that for all j ∈ N , 'j
\(n,n+1)
-bisimilarity is invariant
\(n,n+1)
under i-linking 'i
-bisimilar histories of length
n + 1 in a S5 ETL model. In particular we have for all
g, ga ∈ H that
H k , g 'j
\(n,n+1)
Hk , ga
iff Hk+1 , g 'j
\(n,n+1)
Hk+1 , ga (1)
Moreover it is easy to see that by definition ∼k+1
is
i
still an equivalence relation and that ∼ki ⊆∼k+1
.
Now
i
\(n,n+1)
by IH we have for all g, ga ∈ H that H, g 'j
H, ga iff Hk , g 'j
\(n,n+1)
\(n,n+1)
Hk+1 , ga. Thus by (1) and
transitivity of 'j
-bisimilarity, and since j was
arbitrary the claim follows, concluding the induction
step and the proof.
We can now prove our Main Theorem 16.
Theorem 16. For an S5 epistemic-temporal ETL
model, the following are equivalent:
1. H is isomorphic to asForest(M, P) for some epistemic model M and some state-dependent DEL
protocol P
2. H satisfies PR, NPTP, PT, SG, wUNM, and PS.
of the forest and the properties of the DEL product
update, for any horizontal step h` ∼0i h`+1 there must
also be epistemic accessibilities between all prefixes of
h` and h`+1 of equal length. This means that for each
horizontal step h` ∼0 h`+1 between two histories of
length greater than len(h), we also get h0` ∼0 h0`+1 for
the h-synchronous prefixes h0` and h0`+1 of h` and h`+1 .
This yields a ∼0i -path from h to the h-synchronous prefix of h0 . Since ∼H
i contains the transitive closure of
∼0i , we have shown the claim.
Perfect Tracking. We prove that h ∼H
i he implies
H\(len(h),len(h)+1)
h 'i
he for any h. Assume without
loss of generality that len(h) = n and that h ∼H
i he.
Now assume for contradiction that for all h0 with
0
len(h0 ) = n and h ∼H
i h there is no event f such that
H\(n,n+1) 0
0
0
h f ∼i he and h 'i
h f . By construction, it
means that none of the histories in Ki [h] with length
n were i-connected to a state in Ki [he] by the bisimi0
larity condition. It follows that ∼H
i ∩((Ki [h] ∩ Hn ) ×
0
(Ki [he] ∩ Hn+1 )) = ∅, but this set is also empty after
closure, i.e., ∼H
i ∩((Ki [h] ∩ Hn ) × (Ki [he] ∩ Hn+1 )) = ∅
contradicting our hypothesis. Thus by reduction there
must be some history h0 with len(h0 ) = n and
0
h ∼H
i h
and some event f such that
“1⇒2”: We fix an ETL model
H=
hΣ, H, (∼H
i )i∈N , V
i = asForest(M, P).
By definition of asForest,
we know that
sForest(M, P) = hΣ, H, (∼i )i∈N , V i and ∼H
is
i
the symmetric transitive closure of
∼0i =∼i ∪{(wσ, wσe) ∈ (H × H) | σe ∈ P(w)
sForest(M,P)
and wσ 'i
wσe}. (0)
For any h let Ki0 [h] = {h0 ∈ H | h ∼0i h0 }. We start by
proving that H has the given properties. Propositional
Stability is immediate from our construction.
Perfect recall. Cf. Proposition 14.
0
Synchronous grounding. Assume that h ∼H
i h
0
0
with len(h ) > len(h) for two histories h and h of
the generated forest. If h ≤ h0 , we are done. Otherwise, let ∼0i denote the epistemic accessibility relation of the forest before closing off under transitiv0
ity, i.e., the relation given in (0). Since h ∼H
i h
H
0
and ∼i is the transitive closure of ∼i , we find a sequence of pairwise distinct histories h1 , . . . , hk such
that h = h1 ∼0i . . . ∼0i hk = h0 and for all 1 ≤ ` < k,
either len(h` ) = len(h`+1 ) or h`+1 = h` e` for some e` .
Call the first kind “horizontal steps”. By construction
(2)
and
h0 f ∼H
i he
(3)
H\(n,n+1)
(4)
h0 'i
h0 f.
Since ∼H
i is an equivalence relation it follows from
(2), (3), (4) and transitivity of i-bisimilarity that
H\(n,n+1)
h 'i
he.
No pure time perception.
Assume that
H\(n,n+1)
h 'i
he, then by the bisimilarity clause in
the construction we have immediately h ∼H
i he.
Weak Uniform no Miracles. Let e = (E, e1 ) and
f = (E 0 , f1 ). Assume that there are he and h0 f with
0
H
len(h) = len(h0 ), he ∼H
i h f and h 6∼i he. By Lemma
H
0
15 we have also h ∼i h . By construction we have
S5-closure of the epistemic relations, if we had h ∼H
i
H
H
g ∼H
i ga ∼i he, we would also have h ∼i he, so
∼H
i ∩((Ki [h] ∩ Hn ) × (Ki [he] ∩ Hn+1 )) = ∅. Thus
0
he ∼H
i h f is not obtained by closure, but by product
update and we have E = E 0 . By construction and
definition product update this also implies that
e1 ∼Ei f1 .
(5)
Now assume that g, g 0 , ge, g 0 f ∈ H and
0
g ∼H
i g .
(6)
Now by (5), (6) and product update we have ge ∼H
i
g0 f .
Base Case. Isomorphism for histories in H≤1 × H≤1 is
immediate from the construction.
This finishes the proof of the necessity of our conditions.
Induction step. Assume that we proved everything for
histories in H≤n × H≤n (IH).
“2⇒1”: Given some ETL model
Part A: If he
HETL = hΣ, H, (∼ETL
)i∈N , V i
i
satisfying the stated conditions we show how to construct a matching initial epistemic model and a DEL
protocol. We start with the definition of the initial
epistemic model M0 = hW, (∼DEL
)i∈N , V̂ i:
i
• W := {h ∈ H | len(h) = 1}.
Now we construct the jth event model Ej
hEj , (∼ji )i∈N , prej i:
b iff the following condition holds:
– there exists ha, h0 b ∈ H such that len(h) =
len(h0 ) = j, ha ∼ETL
h0 b and h ∼
6 ETL
ha.
i
i
• Since we shall be using state-dependent protocols,
we can set trivial preconditions prej (e) = > and
therefore shall not need a Bisimulation Invariance
condition.
Finally we update the protocol for each state: if
we1 . . . en ∈ H we add e1 . . . en to P(w). We are now
done with our construction. It gives rise to a forest
asForest(M0 , P) :=
h0 f
If
are
From ETL model to Generated DEL. Assume that
he ∼ETL
h0 f . By Lemma 15 it follows that h ∼ETL
h0 .
i
i
Thus by IH
h ∼DEL
h0
(7)
i
=
• Ej := {e ∈ Σ | there is a history he ∈ H with
len(h) = j}.
(Σ, H, {∼DEL
}i∈N , V
i
∼ETL
i
Case 1: there are histories ge, g 0 f with len(g) =
len(g 0 ) = n, ge ∼ETL
g 0 f and g 6∼ETL
ge, and then
i
i
by construction it follows that
• For every p ∈ prop, V̂ (p) = V (p) ∩ W .
• Set a
0
Now we are in one of two cases.
• Set h ∼DEL
h0 iff h ∼ETL
h0 .
i
i
∼ji
or
he
all
h0 f then he
∼DEL
i
HETL \(n,n+1)
h, h , he, h f are all 'i
-bisimilar.
∼DEL
h0 f then he ∼ETL
h0 f or h, h0 , he, h0 f
i
i
asForest(M0 ,P)\(n,n+1)
'i
-bisimilar.
0
e ∼DEL
f
i
And thus from (7) and (8) by product update he ∼DEL
i
h0 f
Case 2: for all histories g, g 0 with len(g) = len(g 0 ) = n
and ge ∼ETL
g 0 f we have g ∼ETL
ge and from
i
i
0
Lemma 15 it follows that g ∼ETL
g
and
by the fact
i
that ∼ETL
is
an
equivalence
relation,
that
g ∼ETL
i
i
0
g f and by Perfect Tracking that these four histories areETL
bisimilar and that successive in these four
H
\(n,n+1)
are 'i
-bisimilar. This in particular true
0
of h, h , he, h0 f .
From Generated DEL to ETL model. Assume that
he ∼DEL
h0 f
i
h ∼DEL
h0
i
(10)
We are in one of two cases.
Case 1: Either e ∼DEL
f and thus by construction
i
there exists some histories ge and g 0 f such that
h ∼ETL
h0 iff h ∼DEL
h0
i
i
Proof of Claim 23. The proof is by induction on the
maximal length of histories we consider. Given that
the claim holds for all pairs of histories in H≤n × H≤n ,
our induction step works as follows: we prove a similarity claim for pairs in Hn+1 ×Hn+1 (Part A), then we
prove isomorphism for pairs of histories in Hj × Hn+1
with 1 ≤ j < n + 1 (Part B), and finally we prove
isomorphism for pairs in Hn+1 × Hn+1 (Part C).
(9)
By Lemma 15 it follows that
),
Claim 23. Let ∼ETL
be the epistemic relation in the
i
given ETL model. Let ∼DEL
be the epistemic relation
i
in the forest induced over the just constructed epistemic model and DEL protocol. We have:
(8)
len(g) = len(g 0 ) = n
(11)
g 6∼ETL
ge & ge ∼ETL
g0 f
i
i
(12)
and
But then by weak uniform no miracles it follows
from (10), (11), (12) and the fact that len(h) = len(h0 )
that he ∼ETL
h0 f
i
Case 2: e 6∼DEL
f but then by construction and
i
from (9) and (10) it follows that h ∼DEL
he, h ∼DEL
i
i
asForest(M ,P)\(n,n+1)
0
h0 f and these four histories are 'i
asForest(M0 ,P)\(n,n+1)
bisimilar. In particular h 'i
he and
asForest(M0 ,P)\(n,n+1) 0
h0 'i
h f . Concluding this part of
the induction step.
Part B: We prove isomorphism for histories in H≤n ×
Hn+1 .
From ETL model to Generated DEL. Let 1 ≤ j ≤ n.
h0 e. But
Assume h ∈ Hj , h0 e ∈ Hn+1 and h ∼ETL
i
by synchronous grounding there must be a prefix
h∗j ej+1 ∈ Hj+1 of h0 e such that h ∼ETL
h∗j . Since
i
∗
ETL
ETL 0
hj ∼i
h ∼i
h e, with perfect recall it follows
that
EE(h∗j ) ≈ EE(h0 e)
(13)
Now assume for contradiction that a pair intermediate histories immediately succeeding each other between h∗j and h0 e are not epistemically connected, then
EE(h∗j ) 6≈ EE(h0 e), contradicting (13). Thus by reduction all intermediate histories must therefore be epistemically connected, and in particular h∗j ∼ETL
h∗j ej+1
i
∗
ETL 0
∗
0
and hj ej+1 ∼i
h e. If len(hj ej+1 ) < len(h e) we can
iterate the argument. In general we obtain a path of
the form:
h ∼ETL
h∗j ∼ETL
h∗j ej+1 ∼ETL
h∗j ej+1 ej+2 ∼ETL
...
i
i
i
i
∼ETL
h∗j ej+1 ej+2 . . . en−j = h0 ∼ETL
h0 e.
i
i
(14)
By the induction hypothesis, we know that
HETL | ≤ n is isomorphic to asForest(M0 , P)| ≤ n
(15)
0
From (14) we have in particular h0 ∼ETL
h
e.
It
follows
i
HETL \(n,n+1)
then from Perfect Tracking that h0 'i
h0 e. Moreover from Part A, Fact 22 and Observation
20 it follows that
asForest(M0 ,P)\(n,n+1)
h0 'i
h0 e
(16)
But then by construction we have
h0 ∼DEL
h0 e
i
(17)
But from (14), (15) and (17) and the closure condition
of the construction it follows that h ∼DEL
h0 e. Since j
i
was arbitrary, this concludes this direction of this part
of the induction step.
From Generated DEL to ETL model. The first part of
the argument is identical, giving us a path:
h ∼DEL
h∗j ∼DEL
h∗j ej+1 ∼DEL
i
i
i
h∗j ej+1 ej+2 ∼DEL
. . . en−j = h0 ∼DEL
h0 e
i
i
asForest(M ,P)\(n,n+1)
(18)
0
h0 'i
h0 e follows from construction.
From the same argument as in the other direction, it
HETL \(n,n+1)
follows h0 'i
h0 e. Now it follows from no
pure time perception that
h0 ∼ETL
h0 e
i
(19)
By IH we know that the HETL | ≤ n is isomorphic to
asForest(M0 , P)| ≤ n
(20)
From (18), (20), (19) and the fact that ∼ETL
is an
i
equivalence relation, we have h ∼ETL
h0 e. Again j
i
was arbitrary, concluding this direction and this part
of the induction step.
Part C: We prove isomorphism for histories in Hn+1 ×
Hn+1 .
From ETL model to Generated DEL. Assume for
h0 f . Now assume for contradiction
that he ∼ETL
i
DEL
that he 6∼i
h0 f . Then by Part A we have that
HETL \(n,n+1)
h, h0 , he, h0 f are all 'i
-bisimilar. Moreover
by our usual argument we have h ∼ETL
h0 . By induci
tion hypothesis and Part B this is also the case in the
DEL generated forest and by construction and closure
we have he ∼DEL
h0 f . Contradiction. Thus by reduci
tion we have indeed he ∼DEL
h0 f .
i
From Generated DEL to ETL model.
Assume for that he ∼DEL
h0 f . Now assume for coni
ETL
tradiction that he 6∼i
h0 f . Then by Part A we
asForest(M0 ,P)\(n,n+1)
0
0
have that h, h , he, h f are all 'i
bisimilar. Moreover by our usual argument we have
h ∼DEL
h0 and by construction h ∼DEL
he and
i
i
0
DEL
h ∼i
h0 f . By induction hypothesis and Part B
this is also the case in the ETL model. Since this is an
S5 ETL model we have he ∼ETL
h0 f . Contradiction.
i
Thus by reduction we have indeed he ∼ETL
h0 f .
i
This concludes this part of the induction step, the induction step, this direction of the proof and the proof
of the claim.
The sufficiency of our conditions follows from Claim
24.
B
Proof of Theorem 19
“1⇒2”: We start by proving that the epistemic temporal forests generated from an initial model, a clock
tick function and a protocol according to the procedure given in definition have the given properties. By
examination of the proofs given in the preceding representation theorem, it is clear that the preceding properties (except for NPTP) are still true of any model
generated with a clock-ticking function even if one does
not add some epistemic link between two immediately
succeeding bisimilar histories. Below we give, for the
interesting cases, the intuition why properties (except
for NPTP) are preserved).
Perfect recall. The main idea of the soundness
argument (Proposition 14) was to prove inductively
that merging information sets of bisimilar successors (adding ‘vertical’ links) preserves perfect recall.
Forests generated according to clock-tick functions
are intuitively intermediate structures between synchronously DEL generated forests and their asynchronously generated cousins.
Synchronous grounding. ‘Horizontal uncertainties’
still originate by product update and product update
is still applied to synchronous slices.
Perfect Tracking. Clock tick functions allow agents
to distinguish epistemically equivalent states. So it intuitively speaking it allow them to distinguish strictly
more states than in the asynchronous construction, so
perfect tracking is preserved (while NPTP will general
not hold any longer).
Weak Uniform no Miracles. The case of weak uniform no miracles is interesting. The soundness under
clock tick functions stems from the fact that under
Perfect Recall, Synchronous Grounding and S5, the
generated forests will always be isomorphic to one that
could be generated by a well-behaved clock tick function: well-behaved in the sense that it fulfills a form
closure: if wh(E, e) ∼i w0 h0 (E, f ) and i ∈ c(w, h, (E, e))
then i ∈ c(w0 , h0 , (E, f )).
This finishes the proof of the necessity of our conditions.
“2⇒1”: Given some ETL model satisfying the stated
conditions we show how to construct a matching initial
epistemic model, state-dependent DEL protocol and
clock-ticking function. We start with the definition of
the initial epistemic model M0 = hW, (∼DEL
)i∈N , V̂ i:
i
• W := {h ∈ H | len(h) = 1}.
• Set h ∼DEL
h0 iff h ∼ETL
h0 .
i
i
• For every p ∈ prop, V̂ (p) = V (p) ∩ W .
Now we construct the jth event model j
hEj , (∼ji )i∈N , prej i:
=
• Ej := {e ∈ Σ | there is a history he ∈ H with
len(h) = j}.
• Set a ∼ji b iff there are ha, h0 b ∈ H such that
len(h) = len(h) = j and ha ∼ETL
h0 b.
i
• Since we shall be using state-dependent protocols,
we can set trivial preconditions prej (e) = > and
therefore shall not need a Bisimulation Invariance
condition.
We update the protocol for each state: if we1 . . . en ∈
H we add e1 . . . en to P(w). Finally for each triple
(h, σ, hσe) ∈ H1 ×Hj ×Hj+1 our clock ticking function
is defined as i ∈ c (h, σ, (Ej , e)) iff hσ 6∼ETL
hσe.
i
Claim 24. Let ∼ETL
be the epistemic relation in the
i
given ET L model. Let ∼DEL
be the epistemic relai
tion in the forest induced over the just constructed
epistemic model and DEL protocol. We have:
h ∼ETL
h0 iff h ∼DEL
h0
i
i
Proof of Claim 24. The proof follows closely the strategy of the previous theorem, i.e., is by induction on the
maximal length of histories we consider. Given that
the claim holds for all pairs of histories in H≤n × H≤n ,
we prove a similarity claim for pairs in Hn+1 × Hn+1
(Part A), then we prove isomorphism for pairs of histories in Hj × Hn+1 with 1 ≤ j < n + 1 (Part B), and
finally we prove isomorphism for pairs in Hn+1 ×Hn+1
(Part C). To keep this appendix focused on the important changes we try not to repeat arguments when they
are identical in the previous proof.
The base case for histories in H≤1 ×H≤1 and the proof
of part the part A of the induction step follows the
same argument. However we shall prove a stronger
claim (that was not necessary in the proof of the previous theorem):
Part A:
1. If he ∼ETL
h0 f then he ∼DEL
h0 f or we have both
i
i
HETL \(n,n+1)
that h, h0 , he, h0 f are all 'i
-bisimilar
and that they are all ∼ETL
-connected.
i
2. If he ∼DEL
h0 f then he ∼ETL
h0 f
i
i
clickForest(M0 ,P)\(n,n+1)
0
0
or h, h , he, h f are all 'i
0
DEL
bisimilar and moreover h ∼DEL
he
and
h
∼
i
i
h0 f .
Part B: We prove isomorphism for histories in H≤n ×
Hn+1 .
From ETL model to Generated DEL. From the same
argument as in the proof of the previous theorem, we
obtain a path of the form:
h ∼ETL
h∗j ∼ETL
h∗j ej+1 ∼ETL
h∗j ej+1 ej+2 ∼ETL
...
i
i
i
i
∼ETL
h∗j ej+1 ej+2 . . . en−j = h0 ∼ETL
h0 e.
i
i
(21)
By the induction hypothesis, we know that
HETL | ≤ n is isomorphic to clickForest(M0 , P)| ≤ n
(22)
From (21) we have in particular
h0 ∼ETL
h0 e
i
(23)
It follows then from Perfect Tracking that
HETL \(n,n+1)
h0 'i
h0 e. Moreover from Part A, Fact
22 and Observation 20 it follows that
clickForest(M0 ,P)\(n,n+1)
h0 'i
h0 e
(24)
Let σ be the sequence such that h0 = h0 [1]σ. It follows
from (23) by construction of the clock ticking function
that
i 6∈ c(h0 [1], σ, (En+1 , e))
(25)
But then by construction we have
h0 ∼DEL
h0 e
i
(26)
But from (21), (22) and (26) and the closure condition
of the construction it follows that h ∼DEL
h0 e. Since j
i
was arbitrary, this concludes this direction of this part
of the induction step.
From Generated DEL to ETL model. The first part of
the argument is identical, giving us a path:
h ∼DEL
h∗j ∼DEL
h∗j ej+1 ∼DEL
i
i
i
h∗j ej+1 ej+2 ∼DEL
. . . en−j = h0 ∼DEL
h0 e
i
i
(27)
Let σ be the sequence such that h0 = h0 [1]σ. It follows
by construction that
i 6∈ c(h0 [1], σ, (En+1 , e))
(28)
But then by construction of the clock tick function it
follows that
h0 ∼ETL
h0 e
(29)
i
By IH we know that the HETL | ≤ n is isomorphic to
clickForest(M0 , P)| ≤ n
(30)
From (27), (30), (29) and the fact that ∼ETL
is an
i
0
equivalence relation, we have h ∼ETL
h
e.
Again
j
i
was arbitrary, concluding this direction and this part
of the induction step.
Part C: We prove isomorphism for histories in Hn+1 ×
Hn+1 .
From ETL model to Generated DEL. Assume for
that he ∼ETL
h0 f . Now assume for contradiction
i
DEL
that he 6∼i
h0 f . Then by Part A we have that
0
0
h, h , he, h f are all ∼ETL
-connected. By induction hyi
pothesis and Part B this is also the case in the DEL
generated forest. By closure we have he ∼DEL
h0 f .
i
Contradiction. Thus by reduction we have indeed
h0 f .
he ∼DEL
i
From Generated DEL to ETL model.
Assume for that he ∼DEL
h0 f . By our usual argument
i
DEL
0
we have h ∼i
h . Now assume for contradiction
that
he 6∼ETL
h0 f
(31)
i
By Part B and (31) we have h ∼DEL
he and h0 ∼DEL
i
i
0
h f . By induction hypothesis and Part B this is also
the case in the ETL model. Since this is an S5 ETL
h0 f . Contradiction. Thus by
model we have he ∼ETL
i
reduction we have indeed he ∼ETL
h0 f .
i
The sufficiency of our conditions follows from Claim
24.